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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Volume bounds for shadow covering


Authors: Christina Chen, Tanya Khovanova and Daniel A. Klain
Journal: Trans. Amer. Math. Soc. 366 (2014), 1161-1177
MSC (2010): Primary 52A20
DOI: https://doi.org/10.1090/S0002-9947-2013-05855-2
Published electronically: August 20, 2013
MathSciNet review: 3145726
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Abstract:

For $n \geq 2$ a construction is given for a large family of compact convex sets $K$ and $L$ in $\mathbb {R}^n$ such that the orthogonal projection $L_u$ onto the subspace $u^\perp$ contains a translate of the corresponding projection $K_u$ for every direction $u$, while the volumes of $K$ and $L$ satisfy $V_n(K) > V_n(L).$

It is subsequently shown that if the orthogonal projection $L_u$ onto the subspace $u^\perp$ contains a translate of $K_u$ for every direction $u$, then the set $\frac {n}{n-1} L$ contains a translate of $K$. It follows that \[ V_n(K) \leq \left (\frac {n}{n-1}\right )^n V_n(L).\] In particular, we derive a universal constant bound \[ V_n(K) \leq 2.942 V_n(L),\] independent of the dimension $n$ of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.


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Additional Information

Christina Chen
Affiliation: Newton North High School, Newton, Massachusetts 02460

Tanya Khovanova
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Daniel A. Klain
Affiliation: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, Massachusetts 01854
Email: Daniel_{}Klain@uml.edu

Received by editor(s): October 22, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.